A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, and Shuntaro Yamagishi. "A generalization of a theorem of Erdős-Rényi to m-fold sums and differences." Acta Arithmetica 166.1 (2014): 55-67. <http://eudml.org/doc/279059>.

@article{KathrynE2014,
abstract = {Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_\{N\}^\{(m)\}(ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_\{N\}^\{(m)\}(ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.},
author = {Kathryn E. Hare, Shuntaro Yamagishi},
journal = {Acta Arithmetica},
keywords = {sequences; additive number theory; probabilistic methods; thin sets},
language = {eng},
number = {1},
pages = {55-67},
title = {A generalization of a theorem of Erdős-Rényi to m-fold sums and differences},
url = {http://eudml.org/doc/279059},
volume = {166},
year = {2014},
}

TY - JOUR
AU - Kathryn E. Hare
AU - Shuntaro Yamagishi
TI - A generalization of a theorem of Erdős-Rényi to m-fold sums and differences
JO - Acta Arithmetica
PY - 2014
VL - 166
IS - 1
SP - 55
EP - 67
AB - Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_{N}^{(m)}(ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_{N}^{(m)}(ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.
LA - eng
KW - sequences; additive number theory; probabilistic methods; thin sets
UR - http://eudml.org/doc/279059
ER -